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Constructive curves in non-Euclidean planes (slides of a talk)

Presentation · September 2016

DOI: 10.13140/RG.2.2.19257.77927

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Constructive curves in in non-Euclidean pla lanes Ákos G.Horváth

Department of Geometry, Mathematical Institute, Budapest University of Technology and Economics (BME)

19th Scientific-Professional Colloquium on Geometry and Graphics Starigrad-Paklenica, September 5, 2016

Non-Euclidean plane

What does it mean?

A geometric plane distinct from the Euclidean one.

• Spherical plane, hyperbolic plane, projective plane, affine planes with special metric

It is a plane which is not a Euclidean one (With respect to a well-defined synthetic axiom system).

• There are several constructions of „poor planes” which we would not like to consider here.

It is a linear algebraic structure with interesting geometric properties.

• We need preliminary knowledge from some other parts of mathematics.

Connections

Spherical , hyperbolic, Minkowski, Lorentzian

Conic sections. What does it mean?

Two great types of definitions:

• Analytic
• smooth, irreducible algebraic curve of degree 2
• the zero set of a quadratic form above a 2-dimensional vector space
• Synthetic
• By the conic sections (Intersection of a right circular cone with a plane not passing through

the apex of the cone.)

• By some metric property (two focus definition, focus-directrix definition, focus-leading

circle definition)

• By synthetic projective geometry using group theoretical approach.

Results on classification

In the case, when the conic means a smooth algebraic curve of degree two, the problem is clear. With suitable basic transformations we have to give the algebraically non-equivalent canonical forms, than we have to search certain metric definitions of the getting classes. This method gives in the Euclidean case the known three types of proper conics the ellipse the hyperbola and the parabola, respectively. It can be applied in S, H and L.

Spherical conics:

Classification

There is only one class of conics by Dirnböck and Syke, respectively. Syke gave two distinct names for conics lying on a fixed hemisphere (spherical ellipse if it is a closed curve on it or spherical hyperbola if it has two branch on the hemisphere). The metric definition of Altunkaya at all. Immediate gives three types of conics named by ellipse, hyperbola and parabola, respectively.

Hyperbolic conics

Analytic definition and complete characterization Metric definition and characterization Relationship among the definitions

Classification of hyperbolic conics

With respect to a homogeneous coordinate system of the pseudo-Euclidean space the canonical form is The cases with respect to the other two roots of the characteristic equation roots are:

• Th

Ther ere ar are tw two

• dif

ifferent rea eal l roots.

• Coi

Coincidin ing rea eal roots.

• Th

Ther ere ar are tw two

• conj
• njugate complex roots.

Two distinct real roots

Coinciding real roots

Conjugate complex roots

Conics in Lorentzian plane

Metric definition can be found in Analytic definition can be found in

Classification in the Lorentzian planes

Birkhoff used such names as e.g. relativistic ellipse for some curves are defined by certain metric properties. We can find complete classifications in the recent papers:

Conics in Minkowski normed plane

There are only some metric definitions:

Metric definitions for ellipse

Definition with a focus and a leading line

Equivalence of the first two definitions:

Motion of rigid systems in the Euclidean plane (roulettes)

• The plane S’ moving on the fixed plane S
• Fixed frame with coordinates (x,y)
• Moving frame with coordinates (u,v)
• The parameter  is non-vanishing (the motion is

nontranslative)

• For every value of  there is a point of the

moving plane for which the velocity vector

• vanishes. This point (the so-called instantaneous

centre) defines in the moving plane the moving polode (centroid) and in the fixed plane the fixed polode (centroid), respectively.

• The velocity vector at a point is orthogonal to

the position vector from the instantaneous centre to the point, implying that the move is an instantaneous rotation about the instantaneous centre.

Main theorem of planar kinematics and the definition of roulettes

The moving polode rolls without slipping (or without friction) on the fixed polode , and this is the only rolling process which corresponds to the given motion of the planes. Every non-translatory planar motion of a rigid mechanical system in the plane can be considered as the rolling process of a curve (rigidly connected with the system) on a fixed curve in the plane.

The second Euler – Savary equation

Let r and r’ be the curvature radiuses of the fixed and moving polodes at its common point K, and denote by a, the length of the common velocity vector at K. It can be proved with some calculation the second Euler-Savary equation:

The first Euler –Savary equation (geometric form)

The acceleration vector at P can be get as the derivate of the velocity vector at P. The calculation leads to a sum of two vectors, one

• f them is parallel to the normal PK with a

fixed length dependent only on the angular acceleration of the plane around K. The getting other component goes through a fixed point L of the common normal of the polodes, the length of the vector LK is equal to the length of the common velocity vector at K (denoted by a). The Thalesian circle above the segment KL is the circle of

• inflection. If O and I are the curvature centre
• f the roulette at P and the second point of

intersection of the line KP with the circle of inflection we can prove the equality:

Roulettes in the spherical plane

S is the most investigated non-Euclidean plane, since spherical kinematics is an intermediate step between planar and spatial kinematics. There are several paper on it I propose for study the following ones:

Euler-Savary equation

• n the spherical plane

The polodes contact at P and A is the point of the roulette : is the spherical argument of the spherical polar coordinates of A, with the pole-tangent great circle as the spherical 'polar line'. : is a quantity corresponding to the diameter of the inflection circle in planar kinematics, u is the pole changing velocity, and is the angular velocity (magnitude) of the moving body.

Roulettes in the hyperbolic plane

????????????????????? ?????????????????????

Roulettes in Lorentzian plane

• Ergin considering the Lorentzian plane instead of the Euclidean plane, introduced the one-parameter planar

motion in the Lorentzian plane and also gave the relations between both the velocities and accelerations.

• The relations between absolute, relative, sliding velocities (and accelerations) and pole curves was discussed,
• too. In the Lorentz plane Euler-Savary formula is given in references:
• Yüce and Kuruoglu using hyperbolic numbers and reproduced the results of Ergin and in analogy with complex

motions as given by Müller, defined one parameter motions in the Lorentzian plane.

The case of the Minkowski plane

What is the problem with kinematics in Minkowski plane? Busemann notes on it:

Flexible motions in Minkowski plane

Generalized angle measure:

Generalized rotation

Area based general rotation (Kepler’s law)

Second Euler- Savary equation

First Euler-Savary equation

Thank you for your attention!